The straightening/unstraightening equivalence lets us describe functors from an $(\infty,1)$-category $C$ into either the $(\infty,1)$-category of anima $\mathrm{An}$ or of $(\infty,1)$-categories $\mathrm{Cat}_\infty$ as certain fibrations over $C$.
What kind of extra conditions do I need to put on these fibrations to get, for example, finite product preserving functors?
I am mostly interested in the case where $C$ really is the nerve of an ordinary category (like the syntactic category of a Lawvere theory).
